How can I get the position (lat, lon) of a foursquare checkin posted on twitter?

There's actually a much nicer solution here if you would like to use the foursquare API. One that doesn't depend upon a user having enabled Twitter's geo-tagging.

You noticed already that the link in the status update redirects to a foursquare page. Rather than screen scraping, you may notice there are two key pieces of information in the url. The checkin id, and a parameter called s (for signature) that lets you access that checkin via foursquare's API.

https://foursquare.com/dens/checkin/4f0b3676e4b0e8ed3a590e41?s=MWHM3-0BsLXIJprfbALvOM606ZE&ref=tw

For instance, in the above url, the checkin id is 4f0b3676e4b0e8ed3a590e41 while the signature is MWHM3-0BsLXIJprfbALvOM606ZE

Using the /checkins endpoint, you can get the full checkin details (including lat/long of the venue). Dox here: https://developer.foursquare.com/docs/checkins/checkins

You already have what you want, look at the "geo" section in sample of search output you posted. It's null because it is not filled (twitter user forget to enable twitter location in settings and all locations from foursquare are ignored by twitter).

You can filter twits by location, look at the Twitter Dev documentation.

UPD:

If you want to get location even in case when it is not stored to twit, you have to perform request to foursquare short link and parse venue information (you can steal location also). Look at the source of the sample 4sq.com page (small part):

...options['lat'] = 42.98758759825106;options['lng'] = -71.50742411613464;options['fuzzy'] = false;options['venue'] = {"id":"v4f08df964fc62d44946f058a","venue":{"id":"4f08df964fc62d44946f058a","name":"ABC News \u2013 WMUR Debate","contact":{"twitter":"MittRomney"},"location":{"address":"100 St Anselm Drive","lat":42.98758759825106,"lng":-71.50742411613464,"city":"Manchester","state":"NH"},"categories":[{"id":"4bf58dd8d48988d1af941735","name":"College Auditorium","pluralName":"College Auditoriums","shortName":"Auditorium","icon":{"prefix":"https:\/\/foursquare.com\/img\/categories\/education\/default_","sizes":[32,44,64,88,256],"name":".png"},"primary":true}],"verified":false,"stats":{"checkinsCount":7,"usersCount":7,"tipCount":1}}};...

Unfortunately, Foursquare team can change the format of the page or limit requests from IP Address without any notes. That's why this solution is ugly.

UPD2

See answer from @MikeLewis, I've missed that Foursquare API supports requesting information by checking id.